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Intermediate structure and the reaction 28Si(p,pt́)28Si
Nuclear Physics A159
(1970) 577--597; ( ~
North-Holland Publishing Co., Amsterdam
N o t to be reproduced by photoprint or microfilm without written permission from the publisher
I N T E R M E D I A T E STRUCTURE AND T H E REACTION
2SSi(p, p')ZSSi* A. C. SHOTTER, P. S. FISHER and D. K. SCOTT
Nuclear Physics Laboratory, Oxford Received 27 July 1970 Abstract: Angular distributions are reported for the reaction 2sSi(p, p')2sSi* measured over t h e incident energy range 12.0 to 15.0 MeV, at 100 keV intervals with a beam resolution of 100 keV. Total cross sections are determined for all residual states from 0.0 to 8.25 MeV. Cross-correlation analysis between the cross sections reveals structure similarities between the different excitation functions; this structure has a width of several hundred keV. An attempt is made to interpret the structure within the framework of the conventional statistical model. When the spacing of the compound levels is described by a Poisson distribution the model gives a fairly good representation of the data.
E I
NUCLEAR REACTIONS "sSi(p'P)' E=12"0-15"0 MeV; measured a(E;Ep;'O)"
]
29p deduced intermediate structure. Natural target.
1. Introduction In the past few years much interest has been focussed on the occurrence of intermediate structure in the excitation functions for nuclear cross sections and on the interpretation of the structure using the concept of doorway states 1,2). Although many experimental examples of intermediate structure have been reported, only one reaction channel is studied in many of these cases, and since it has been shown 3, 4) that the occurrence of intermediate structure is not alien to the statistical theory of nuclear reactions, some caution must be exercised when considering the interpretation of the data. In contrast to the conventional statistical theory, however, the doorway state theory predicts, in general, that similar intermediate structure should occur in all open channels 2). It is therefore of considerable interest to measure excitation functions corresponding to several inelastic channels, and then to study the correlation in structure between these functions. To this end, we have measured inelastic proton excitation functions for the nucleus 2SSi. This nucleus was chosen because it belongs to a mass region in which the total neutron cross sections, in the energy range of 4 to 12 MeV, undergo considerable fluctuations 5), and furthermore 28Si is unique in this mass region in having a doubly closed subshell, which would simplify calculations of the reaction mechanism. The reaction was investigated in the incident proton energy range 12 to 15 MeV, corresponding to an excitation in the compound nucleus of between 14.7 and 17.7 577
578
A.c.
SHOa'TER e t al.
MeV. Since'the purpose of the experiment was to search for intermediate structure, it was decided to reduce the amount of data to be analysed by measuring the reaction cross sections averaged over an incident energy interval of 100 keV. 2. Data collection and reduction The Van de Graaff accelerators of the Oxford Nuclear Physics Laboratory were used for this experiment 6). Details of the technique employed to measure energyaveraged cross sections are given elsewhere 7). Briefly, the method consisted of cyclically changing the incident beam energy by modulating the current energising the final analysing magnet in a saw-tooth manner with a period of 100 see. The deterioration of detector resolution that would have resulted from this modulation of the beam energy was avoided by electronically compensating the pulse-height spectrum of the inelastically scattered prgtons. In this way we found it possible to have a beam resolution of several hundred keV while maintaining an overall energy resolution between 25 and 35 keV in the inelastic proton spectra. ,
,
li:o
t
•
,
,
Peak. State (MeV)
0-0 Si
2 d
7
13 14
oo o
0.0 C
,
,
,
r
i
I
I
Peak. State (MeV) I I m I l I
16 18 19 20 11
8.16 Si 8.33 si 8"41Si 8'59 Si 8'908i 7.66C ~945
1.77 5J 4"61 Si 4"97 Sl 4"43C 6"1 7 5i 6"68 Si 6"88 S116-130 7.38,7.41 Si
I
22
9-17 Si
I [ i ~
23 14 28 26
9"31Si 9"dl 51 11"880 9'70 Si
6.920
I
27
7-80 Si;7.12 0 I 7" 93 5i
28 19
9.765i
9-93 Si 10"185i;9.590
J t.d Z Z
"1" tJ
0
400 CHANNEL Fig.
1. A
800
spectrum of scattered protons from the silicon target.
The angular distributions of the scattered protons were measured by simultaneously recording the pulses from four Si(Li) detectors mounted on a rotating table in a 60 cm diameter scattering chamber. The detectors, which had depletion depths of 2
2sSi(p, p')2SSi*
579
mm, were cooled to liquid nitrogen temperature to improve their resolution. The cross sections at sixteen angles could be obtained by taking data at four positions of the rotating table. The targets used for this experiment were self-supporting, 100/zg/cm 2 silicon films made by evaporation of natural silicon. A typical spectrum of scattered protons from such a silicon target is shown in fig. 1 and it can be seen that there is considerable carbon and oxygen contamination. During the course of the experiment about one thousand pulse-height spectra were collected; each spectrum consisted of one thousand channels. The spectra were analysed by a computer programme which included the following procedures: a search routine to determine the regions containing isolated or overlapping peaks; a routine to calculate the background, in regions containing peaks, by interpolating between peak-free regions; an iteration procedure to fit a Gaussian line shape to each peak located in the reduced spectra produced by subtracting the background; a kinematic routine to identify the peaks. The method of least-squares was used to fit each experimental angular distribution with an expression of the form do"
Im~z -
d12
-
~ BtP,(cos 0).
(1)
t=o
The integrated cross section was then calculated as 4roB0. 3. The experimental results The total cross sections for inelastic proton scattering to the first eleven states of zssi are shown in fig. 2. The experimental angular distributions change rapidly with energy, some indication of which can be appreciated from the plot of the Legendre coefficients in fig. 3. From fig. 2 it can be seen that the integrated cross sections exhibit a structure which has a width of several hundred keV; this structure is termed intermediate structure since its width is intermediate between the widths observed in "fluctuation" experiments and the potential resonances that are seen with beams of poor energy resolution. It is also apparent from fig. 2 that there is some similarity between the structures of the different excitation functions. In order to establish a quantitative level of significance for the similarity of any two excitation functions, cross-correlation functions were determined for all combinations of the cross sections. For any two cross sections tri(E), trj (E) the cross-correlation function, Rij (e), is defined by
Rij(e)
= ((o-,(E)-
)(o-j(E+ e)<~j>
where (a(E)> is the energy average of o-(E).
)>,
(2)
580
A.c. SHOTTER et aL
The cross-correlation functions for all combinations of the cross sections are shown in fig. 4. The uncertainties of R u (s), which are caused by the finite range of the experimental data, are represented by an error bar plotted at the side of each graph in Q = 0.0 HeY
40T
2000 l I0 O0t . -'% ~..,,,........,."" "%% ,,,",,~,,,...,..-"% 13 ENERGY
14 15 MEV o - 1.77 McV
z°° T/
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/ER- 3%
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=.~ ~
*"'-",../ %
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"~'%,.~'. 14
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IS
| SUM OF QI FROM 1"77, TO 7.93 McV 12 13 14 15 LAB ENERGY MeV
Fig. 2. The integrated cross sections for the reaction 2ssi(p, p')Zssi*. The excitation energy o f the residual nucleus is denoted by Q, and the approximate percentage error on the experimental points is given by ER.
fig. 4. These errors were calculated from the formula derived by Hall 8), modified to take account of the finite energy resolution of the beam used in this experiment. If the different reaction cross sections are unrelated in energy, then in any one graph of fig. 4 there should be approximately the same number of points lying within the error bar as outside since the error bars correspond to the most probable errors. The
s SSi(p, P')2SSi*
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582
A.C. $HOTTER et aL
interpretation of the behaviour of R~s (s) for the graphs involving the excitation functions of the lower excited states of 28Si is complicated by the possible contribution of a direct component to the reaction mechanism; this will cause the value of R~s (~) E~ ~ATE ~
0"0 MeV
Ci I-~ ~V
4.61 MeV
~ 4.97 McV
• ..t_~o2
6.Z7 MeV
• .~L~;,
E~
E~
7.4 MeV
GSB MeV
7"8 MtV
t_~o,
7"9 McV
l -O'OZ
...+ • ..+ ÷ ..+.. + + . °.
•
+++ ....+..+ i
\
Fig. 4. The cross-correlation functions. Each c o l u m n consists o f graphs s h o w i n g the correlation between one particular reaction cross section and the other remaining cross sections. The cross sections are identified by the excitation energy o f the residual nucleus. The y-axis scale is the same for all graphs in any c o l u m n , and any t w o neighbouring points in any one graph are separated by e = 100 keV.
to be depressed as can be seen from eq. 2. Nevertheless, there are at least six different reaction channels for which the cross sections, when correlated with the cross section for the 1.77 MeV state, yield values of R, over an interval of 300 keV, greater than the errors due to the finite range of data.
2SSi(p, P')28Si*
583
Of the forty possible different combinations of the excitation functions only six cases have Rij (8) negative for 8 = 0. If the different reaction cross sections are unrelated in energy, and if the data are of infinite extent, then the value of R~j (8) would be zero. However a finite value of Rij (8) can arise due to the limited range of the data, and in such a case both positive and negative values of R~j (8) will occur with equal probability. The fact that so many experimental values of Rij (e = 0) are positive, and comparable in magnitude with the value of the autocorrelation function Rii(8 = 0), is indicative of some correlation between the reaction cross sections. If the states at 6.27 and 6.88 MeV, for which R~j fluctuates about zero, are excluded, the values of R~j (~) are positive for a region of several hundred keV centering around 8 = 0 and, for many of these cases, the values within this region are consistently greater than would arise by chance because of the finite range of data. A clear illustration of these correlations in the cross sections is seen in fig. 2, which in addition to showing the experimental excitation functions for individual states, shows the sum of cross sections for scattering to groups of states. It is clear that there is strong similarity between the structures of the single and summed excitation functions. 4. The statistical theory and intermediate structure The excitation functions shown in fig. 2 exhibit intermediate structure, and there appears to be some correlation between the structure for the different reactions. The question as to whether doorway states are responsible for the structure still remains to be answered. As noted by previous authors 3) statistical grouping of the compound nuclear parameters could be responsible for the structure. That this may indeed be so can be appreciated from the following simple argument. Consider a reaction which proceeds by a compound nuclear mechanism. At low bombarding energies discrete resonances are observed, whereas at higher bombarding energies the average energy spacing ( D ) between the resonances decreases rapidly, while the average width ( F ) of the resonances increases only slowly. For the energy region where there is strong overlapping between neighbouring resonances the cross sections become quite complicated, and for the region where (F)/(D)>> 1 the "resonances" or fluctuations in the cross section can no longer be identified with individual compound nuclear levels but are rather the result of interference between the amplitudes of many levels. The position and amplitude of these "resonances" are essentially random. Since many levels contribute to the reaction when (F)/(D) >> 1, and since the decay amplitudes to the different reactions channels are in general independent, the cross sections for different inelastic channels will in general be uncorrelated. This behaviour is to be contrasted with that in the lower energy region since in general an isolated resonance will decay to all open channels, and so there will be strong correlation between the different inelastic cross sections. In the intermediate energy region, where (F)/(D) ,~ 1, some correlation is still likely to exist between the inelastic cross sections.
A . c . SHOTTER et aL
584
For the experimental situation we have studied, (F)/(D) ~ 2. It has therefore been necessary to estimate the correlation between the reaction channels for this intermediate energy region before concluding that doorway states are the cause of structure correlations between different reaction cross sections. In the next section we present some details of the method we have used to calculate the statistical correlation when (F)/(D) ~ 1. Our notation and method are similar to those previously used by Ericson 9). 5. The statistical model
The cross section for a reaction between two states ~ and ~' may be written in the following form 10)
=
E(2t+
2.1+ 1
iS(aisl ,l,s,ljn)12'
(3)
1)(2i+ 1)
where S (adslat'l's'lJn) is the scattering matrix element describing the transition from the state • of channel spin s to a state r~' of channel spin s', through a compound system with a total angular momentum J and parity n, the incident particle having an orbital angular momentum 1, and the emergent particle an orbital angular momentum l'. The spin of the target nucleus i s / , and the spin of the incident particle is i; the sum is over the quantum numbers 1, l', s, s', J, n. It is assumed that the scattering matrix elements may be separated into two components, one associated with the compound nuclear mechanism and the other with a direct reaction mechanism 1i), i.e.
S(~lslat'l's'lJn) = ~ ~ - E , ~ - ~ +(S),
(4)
where the summation extends over all the compound nuclear levels associated with the quantum numbers J and r~; ( S ) is the energy-averaged component associated with a direct reaction mechanism; the term E~ = ei-½iF~, where ei and F~ are the energy position and width of the compound nuclear level i; the index/~ represents the quantum numbers/, s, 1', s'. For the purpose of this work it will be assumed that the real and imaginary parts o f the coe~cients a l ( ~ ' , / ~ J n ) belong to Gaussian distributions, and that the coefficients corresponding to a particular i, but to different values of (~a',/~Jn), are independent. With further assumptions about the distribution of the compound nuclear energy positions it is possible to calculate energy-averaged cross sections and the fluctuation from the mean. A measure of the local energy fluctuation of a cross section is given by the variance o f the cross section defined by var (a) = =
((a-(a)) z)
where the angle brackets denote an average over energy.
(5)
2aSi(p,P')iSSi*
585
For simplicity the direct reaction component of the scattering matrix is assumed to be zero; in this case
a(~l~')
ai(~tot',flJn)a*(otot',flJn) g(J) ~', .h,ts U (E- Ei(Jn))(E- E~(Jn))
= Y.
t6)
where
g(j)
=
nA2(2J+l)
(21+0(2i+0 .Using this equation the averaged square of the cross section may be written
<~2(=1=')> = E g(d)g(J') ,lx# ,,l'n'B'
a,(flJn)a~([JJ~c)am([3'J'rg)a*(fl'J'n') \ × ,~. (E- Ei(Jrc))(E- Ej(Jrc))*(E- E.(J n'))(E- E.(J'n'))*I"
(7)
When the random nature of the coefficients is considered, the sum of the terms in this expression for which J = J', n = n', fl = fl' becomes Eg2(J) I2 ( ~
.r~p
fa'[2lajl2\ I(E-~Ej)I2/,,j
ie-e,i /
\,, ( e - ~ E * ) ' / , + , / "
(s)
The values of the first and last terms in this expression depend upon the correlation between the energies of neighbouring compound nuclear levels. If the levels of particular J and n are randomly distributed along the real energy axis, then expression (8) becomes x2) la'14 . \ l E F)~.,-I-g2(j)/V. J,,a L\ " ; IE-E,14/_I " For the case where the average compound nuclear level width is greater than the average level spacing, the summation in this expression may be approximated by an integration over E, with the result that the above expression becomes
E I22-#+ 8/tg2(j) <1a+12>21, ,.p D(Jrc)F(Jn)3
(9)
where D(Jn) is the average level energy spacing for states with the quantum numbers J and n, and F(Jn) is the average total width for the compound levels.
A . C . SHOTTER e t al.
586
By using "similar procedures to those just outlined, the average cross section may be written (a(~[~')) = ~ F ( ~ n ) ( l a ' 1 2 ) 2
(10)
and so expression (9) becomes )-" 2(a(~l~'))s2.p[1
+O(Jn)/nF(Jn)].
(11)
drip
The evaluation of the terms of eq. (7) for which one of the equalities J = J', n = n', p = ~' does not hold, follows along similar lines as the evaluation of expression (11 ); the result is
'
'
F hpp,6ss, 6.., [_nF(Jn)/D(Jn)
(a(~[~))s.p(a(~[~ ))s'.'p'
]
+1 ,
(12)
J,x,p,
where hpp, = I if/~ #/V, hpp, = 0 if p = p', and 6 is the Kronecker symbol. After collecting terms for eq. (5), the variance of the cross section becomes i
var (a(~[~')) = Z (a(~l~'))s2.p [1 + D(Jn)1 s.p ~F(Jn)]
2
~r(Jn)/D(Jn)
(13)
When the cross section includes scattering to several residual states, so that a=Za(~[~,) ,
(14)
n
the variance can be evaluated by similar methods to those outlined above, with the result var (a) = E (a(~l~.))~s. (1+ D(Jn) ~ .ps. \ ~r(Jn)/
+
cr(~[~n)#sg)2 #n
nr(Jn)/D(J~)
(15)
6. Numerical calculations The validity of eq. (15) was tested by simulating excitation functions from preset parameter distributions of the energies and the amplitudes of the compound states. The starting point of these calculations was the S-matrix
S(E)
= ~
a,
E~-E~
= ~
~,+i]~,
• E-Ei+½iF'
(16)
where it is assumed that the real and imaginary parts of the reaction amplitude as belong to Gaussian distributions of the same dispersion, but are otherwise unrelated.
2asi(p, p')2aSi*
587
We investigated excitation functions characterized by three types of level spacing distributions; constant level spacing, a Poisson distribution of level spacing and a Wigner 13) distribution of level spacing which corresponds to a level repulsion. The random numbers used for these calculations were generated as a pseudorandom sequence 14) by the congruential method using a module of 515. The excitation functions were generated by calculating the expression tr(E) oc Y'
(a,+ ifl,)(~j+ iflj)*
(17)
77( E - E, + ½ir)(E- E j - ½iF) as a function of E. At each energy E, the summation in this equation should have extended over all resonances i, but in practice only states within an energy interval of the order F made any significant contribution. For the calculations to data we have used a summation interval of 10F, i.e. for a particular value of E only those levels which occur in the interval E - 5 F to E + 5F are considered. TABLE I Variance o f artificial excitation functions <_P)/
Average n u m e r i c a l variance
Theoretical variance
c o n s t a n t spacing
0.126
l/n = ~- = 0.125; fluctuation theory
r a n d o m spacing
0.390
0.4; calculated from eq. (15)
W i g n e r spacing
0.240
The ratio of the terms occurring on the right-hand side of eq. (15) depends upon the number of reaction channels n. Therefore in order to enhance the contribution of the second term we have studied excitation functions comprising eight independent channels. Since these calculations require considerable computing time, we have so far considered only excitation functions parameterized by F/D = 2, which is closest to our experimental situation. The average numerical variances for the excitation functions associated with the three types of level distributions mentioned above are shown in table 1, together with the value predicted from conventional fluctuation theory. It is seen that the variance for excitation functions characterized by a Poisson distribution is about three times greater than the value given by fluctuation theory, but, as shown in this table, the variance is in reasonable agreement with the value calculated from eq. (15). As expected, the variance of the excitation functions characterized by a Wigner spacing is substantially smaller than the value for the case of Poisson spacing, while the variance of excitation functions with constant level spacing agrees with the value predicted by fluctuation theory 9).
:588
A.C.
SHO'I'rER
et
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590
A . C . SHOTI*ER e t al.
The grea~er variance of the simulated excitation functions for random level spacing compared with constant level spacing is primarily due to a grouping of the levels, which produces similarities of structure in the different excitation functions. This point is illustrated in fig. 5 where three different excitation functions belonging to the same random sequence of energy levels are shown; there is an obvious similarity between the curves. The three excitation functions displayed in fig. 6 correspond to a constant level spacing, and in contrast to fig. 5, there is no similarity between these curves. 7. The experimental results and the statistical model In this section we outline the method and approximations used to evaluate the cross-section variances from eq. (15) and we then compare these calculated values with the experimental variances. The average cross sections appearing in eq. (I 5) for the Variance of the crosssections are for specific values of J and ~ in the compound system. These cross sections were calculated using the formalism of Hauser and Feshbach 15). The relevant transmission coefficients, which are required by the formalism, were taken from the work of Percy 16) and the ratios F(JTt)/D(Jn) were calculated by means of the statistical model 17) which gives 1
F(Jz)/D(Jn) = ~ ~
Q(IJrOf p(E*,)T~(E,)dE,,
(18)
where Q(lJn) = ~f(jTt), a n d f ( j ~ ) is the spin distribution function for the residual nucleus. In this equation r denotes a particular decay mode e.g. p or 'tHe, p(E*) represents the total density of levels for the appropriate residual nucleus at excitation energy E*. The transmission coefficient for an emitted particle of spin I decaying from a compound nuclear state with quantum numbers J~, with orbital angular momentum l, and energy E,, is denoted by 7", (E~). To simplify the evaluation of eq. (18) the dependence of p upon E was taken to be 18) p = exp (E-E°-----~)
(19)
7"0 and the spin distribution function to be 19) J (jrt) = (2j + 1) exp ( - (j + ½)2/2a2). 2tr 2
(20)
The values of the constants Eo, To and tr were taken from the work of Gilbert and Cameron 20). When the average cross sections, determined by means of the above formula, were summed with respect to J and 7t the values were less than the corresponding average experimental cross sections by a factor of 1.6. This underestimation is probably due to an incorrect estimation of the absolute magnitudes of the ratios F(Jn)/D(Jn) calculated from eq. (18), and so, in subsequent work, the calculated average cross sections were multiplied by a factor of 1.6.
2Ssi(p, p')28Si*
591
The variances of the cross sections calculated from eq. (15) were multiplied by a damping factor to take account of the finite energy spread of the incident proton beam; this factor was deduced from the work of Van der Woude 21). The calculated variances are compared with the experimental values in fig. 7; it is seen that within the experimental errors there is approximate agreement between theory and experiment.
/
8O E A
6C
I
I
STATES I. Q " 1.77 MeV 2. Q'4.61 McV 3. Q "4"97 McV 4. Q - 6.27 McV 5. Q " 6'88+6.89 MeV 6. Q-7-38+7.41 McV 7. Q -7.80 McV 8. 0 "7'93 MeV
g
2C
I
I
I
I
20
I
40
I
I
I
I
I
60 80 I00 EXPERIMENTAL VAR (o) mb2
I
I
I
120
I
140
Fig. 7. The calculated and experimental variances of the integrated cross sections for the reaction 28Si(p, p')28Si*. The different cross sections are identified by the corresponding energy of the residual nucleus. The line represents the equation y = x.
In our investigation of the correlation in structure between the experimental excitation functions we have found it more advantageous to study the variance of excitation functions produced by compounding several of the experimental functions, rather than to study the correlation between the individual functions. The reason for this may be appreciated from the following argument. If we suppose that the inelastic cross sections are uncorrelated in energy, then the variance of a compounded excitation function should be given by the equation var
(E a(~l~.)) = E var (a(~]=.)), n
(21)
n
where n labels a particular reaction channel. If, however, eq. (15) is correct eq. (21) must be incorrect, and in this case should be replaced by
(E o(=l=.),p>(E var
(E o(~1~.)) = ~ var (o(~l~.)) + Z n
n
Jxlln'
P
#
nl"(J=)/O(Jg)
(22)
From this formula it is evident that the ratio of the second to the first term is proportional to n, the number of channels, and so structure correlations between the indi-
592
A.C. SHOTTER et al.
vidual excitation functions become more apparent in compounded functions. It is worth noting that eq. (22) reduces to eq. (21) as the ratio F(J~)/D(JTr)tends to infinity. In fig. 8 the variances of several experimental compounded excitation functions are plotted against the values calculated from eq. 21; it is evident that the experimental values are greater than the values predicted from eq. (21) - a result indicative of d(R) SUM OF CROSS SECTIONS FOR / STATES WITH Q VALUES: / 4-61, 4.97, &Z7, 6.88+, 7.41",7 - ~
Nt 3°°
/
.~ Z00
•
7 :
• • Z. . . . 3. 4. 5.-
/ c~
/ IOC
,,f
/
/
6.-
7. ,,
-vor(o (R)-o (dF61)) vor(d (R) -o (6"27)) = mr(¢ (R) -o (6.88)) - vor(o (R)-o (7"4)) - vor (o (R)-el n'-B)) - w r (e (R)-e (T.9)) - vor(o (R))
200 300 i 400 EXPERIMENTAL VAR (~) mb2
t 500
Fig. 8. T h e calculated a n d experimental variances o f several c o m p o u n d e d cross sections for t h e reaction zsSi(p, p')ZsSi*. T h e n u m b e r n o t a t i o n identifies different c o m b i n a t i o n s o f the experimental cross sections. T h e calculated variances were evaluated u s i n g eq. (21). T h e line represents t h e e q u a t i o n y = x. .j
7
400 ! 3O0
~ZOO
i
NUMBER NOT AS FOR FIGAsTION
I00 /
'
300 400 EXPERIMENTAL VAR (or) rnbz
500
Fig. 9. The calculated and experimental variances of the compounded cross sections for the reaction zsSi(p, p')ZsSi*. T h e calculated values were evaluated using eq. (22). T h e line represents the equation y ~ .*¢.
2SSi(p, P')28Si*
593
structure correlation. Fig. 9 is similar to fig. 8 except that the calculated values were derived from eq. (22); clearly, for this case, there is better agreement between the calculated and experimental values. The next two sections deal with the experimental angular distributions, and in particular with their dependence upon energy. 8. The Legendre coefficients The angular distributions may be parameterised by a set of coefficients BL, defined by the equation [ref. lo)] do - ~ B,(ulu')Pr.(cos 0). dO L
(23)
The coefficients B, are related to the scattering matrix by the equation X2 BL(Ctl*t') = 4(2• + 1)(2i + 1) ~ ( - 1)"-SZ(ll J112 J2; × Re
sL)Z(l~ Jl 1'2J2; s'L)
[S(*tll sl~t'l l't s'lJx nx)S*(ot/2 slot'l'2s'[J27t2)'] ~t2
4(2• + 1)(2/+ l) L~ + E w(cl, c2, L) Re
~(c,
¢,
L)S~,(c)
[S,=,(cl)S*,(c2)]],
(24)
ClC2
cl -~c2
where cl is an abbreviation for the quantum numbers 11, I~, s, s', Jx, n~ and = (ct, c2) = (-1)~'-*Z(ltJll2J2; sL)Z(I~Jll~J2; s'L); otherwise the notation is the same as for eq. (3). The first term in eq. (24) is non-zero only for even L coefficients; the second term may be separated into two components, depending on (cx, c2), one associated with odd coefficients, the other with even coefficients. Since the same combinations of (ct, c2) appear in all the coefficients with L even, it is probable that these coefficients will exhibit some correlation. Similar conclusions hold for the coefficients with L odd. It has been further shown 9) that, if the direct component of the scattering matrix is zero and if the compound nuclear component is completely independent for the different channels c, then there is zero cross-correlation between the odd and even coefficients BL. The experimental values of the coefficients BL, which are shown in fig. 3, are not inconsistent with these conclusions, since there is some evidence of correlation between the coefficients with L even and also between the coefficients with L odd. The effect is particularly marked for inelastic scattering to the 4.97 MeV state of 2SSi; furthermore for this state there is little or no correlation between the even and odd L coefficients.
594
A . c . SHOa'rER et al.
9. Variance of Legendre coefficients By using the same assumptions that were employed in the derivation of eq. (15) it can be shown ,2) that the variance of the coefficient BL is related to the partial reaction cross sections a(flJn) in the following manner var (BL(Ula')) = ~ w2(c, c, L)(a(c))2 + ~ w2(cl , c2, L)(a(Cl))((7(c2)) Cl, ¢2 C l ~= C2
C
1
+ c,~',c, nF(Jrc)/O(Jn)
[w(cx, cx, L)w(c2, c2, L)+w2(c,, c2, L)], (25)
where c = (fl, J, n), fl = (l, s, l', s') and Y" denotes a summation with the restriction J, = J2 = J, rq = n2 = re. The coefficient w(cD cz, L) is defined by
w(c,, c2, L) = z'(ll, a,, h,
s , , t;, s,; s'L)
4n~/(2J 1 + 1)(ZJ2 + 1) Comparison of the terms in eq. (25) with those of eq. (15), in association with eq. (24), suggests they are similar in origin. The first term of eq. (25) is analogous to the first term of eq. (15) and arises mainly from interference between the compound resonances associated with a particular set of quantum numbers c. The second term in eq. (25) arises from an interference between compound resonances associated with different sets of the quantum numbers c; this interference is also represented by the second term in eq. (24). The third term in eq. (25) is analogous to the second term of eq. (15), and arises from the contribution of the single resonance states to the variance; this term becomes negligible when F/D >> 1. TABLE 2 Value o f term in eq. (25) (mb/sr) 2 L
1st term
1 2
3rd term
95 29
3 4
2nd term
104
7
134 14.2
124
3.3
We have applied eq. (25) specifically to the case of inelastic scattering to the 4.97 MeV state of 2aSi, which has spin 0 and therefore makes the evaluation simple. The calculated values of the terms occurring in eq. 25 for this reaction channel are tabulated in table 2. It is clearly seen that the value of the second term, due to interference between levels with different J and n, dominates the other terms. The calculated and experimental values of the variances of the coefficients BL are plotted in fig. 10. There
2SSi(p ' p,)ZSSi*
595
is reasonable agreement, although the large experimental errors make it difficult to draw any firm conclusions. 4C ;
.(3
•
l
I
3c ..J n,-
i
•
IL VALUES 0. I. 2. 3. 4.
J
~ 1c J < U
I
I I 20 EXPERIMENTAL
I I I 40 6O VAR (B L) ( m b / s r ) 2
L'0 L-I L-2 L-] L'4 I
I
80
Fig. 10. The calculated and experimental values of the variances for the Legendre coefficientsof the reaction zssi(p, p')Zssi*, Q = 4.97 MeV. The line represents the equation y = x.
10. Discussion and conclusions The motivation for the experiment outlined in this paper was the proposal that intermediate structure may occur in reaction cross sections, and that the structure associated with different reaction channels may be correlated with energy. We have measured excitation functions for several channels of the reaction ~aSi(p, p')2aSi*, and we have used the cross-correlation function to determine if there is any similarity between these functions. The results of these studies give evidence for structure correlation between the excitation functions, which in the past has been taken as convincing evidence for the existence of doorway states. However, before identifying this intermediate structure with doorway states we have studied the conventional statistical model in order to see if this can account for our experimental results, in particular for the similarity of excitation functions for different reaction channels. The picture of intermediate structure that emerges from the statistical model is that of a grouping or clustering of compound nuclear levels at particular energies; the grouping arises because the energy positions of the compound nuclear levels are essentially random. This grouping of the compound nuclear levels does of course influence the cross sections for all open channels, and so tends to produce similar structure in all reaction cross sections. The degree of correlation between cross sections depends upon the magnitudes of the ratios F(J:t)/D(Jn); for values much greater than unity the correlation will be very small, but, for values around our experimental value of 2, the correlation becomes significant. If it is assumed that the energy positions of the compound levels are described by a Poisson distribution along the energy axis, then the grouping of the compound levels gives rise to correlation similar to that ob-
596
^. c. SHOTTERet aL
served experimentally. If, however, the distribution of the le,vel spacing is of the Wigner type 13), then the levels will be more evenly spaced than in the previous case, the grouping of the levels will be less pronounced, and so the correlation between reaction excitation functions will be smaller. The results of the numerical calculations discussed in sect. 6 lead to the conclusion that if there is a Wigner distribution then the grouping of the compound levels will be too insignificant to explain our experimental results. There is considerable experimental evidence 22) to show that the level spacing distribution at low excitation energies is of the type proposed by Wigner. At higher energies where the levels overlap the situation is far from clear, but it is usually assumed that the mathematical form of the level distribution is similar to that for low energies. However, recent calculations by Moldauer 23) suggest that this may not always be true. In these calculations resonance pole parameters of the scattering matrix are calculated from statistical models of the K-matrix, so ensuring the unitary character of the scattering matrix. The results of these calculations clearly show that the repulsion between levels decreases when the channel transmission coefficients and the number of open channels increase. F o r the experimental situation we have studied, several of the transmission coefficients are in fact quite large and the number of open channels is high, and so from Moldauer's results it might be concluded that the compound levels of particular J and n show little repulsion. I f this is true, then the levels are randomly spaced, and this is exactly what is needed to support the interpretation that the experimental structure correlation is largely a statistical effect. Further experimental data are clearly required before forming definite conclusions about the observed intermediate structure. One possible experiment would be similar to the one set out in this paper but for a heavier target nucleus or a higher bombarding energy so that F ( J n ) / D ( J n ) >> 1. F o r this case a conventional statistical interpretation of any observed similarity between the excitation functions would be untenable. We are grateful to Professor D. H. Wilkinson and Professor K. W. Allen for the use of the Oxford electrostatic generators. We wish to thank the technical staff of this laboratory for their willing support. A.C.S. and D. K. S. wish to thank the Science Research Council for financial support.
References 1) H. Feshbach, Int. Conf. on the study of nuclear structure with neutrons, ed. M. N~ve de MCverghies et aL (North-Holland, Amsterdam, 1966) 257 2) H. Feshbach, A. K. Kerman and R. H. Lemmer, Ann. of Phys. 41 (1967) 230 3) P. P. Singh, P. Hoffman-Pinther and D. W. Lang, Phys. Lett. 23 (1966) 255 4) W. R. Gibbs, Phys. Rev. 181 (1969) 1414 5) K. Tsukada and O. Tanaka, L Phys. Soc. Jap. 18 (1963) 610; U. Fasoli, D. Toniolo, G. Zago and F. Fabriani, Nuovo Cim. 44B (1966) 455; A. D. Carlson and H. H. Barschall, Int. Conf. on the study of nuclear structure with neutrons, ed. M. NCve de MCvergnies et aL (North-Holland, Amsterdam, 1966) 537
zsSi(P, p')2s$i* 6) 7) 8) 9) 10)
597
W. D. Allen and R. H. V. M. Dawton, Rutherford lab. report RHEL/R/120 (1966) A. C. Shorter, J. Takacs and P. S. Fisher, Nucl. Instr. to be published I. Hall, Phys. Lett. 10 (1964) 199 T. Ericson, Ann. of Phys. 23 (1963) 390 H. Feshbach, Nuclear spectroscopy, Part B, ed. F. Ajzenberg-Selove (Academic Press, New York and London, 1960) 625 11) H. Feshbach, Ann. of Phys. 43 (1967) 410 12) A. C. Shorter, D.Phil. thesis, University of Oxford (1968) 13) E. P. Wigner, Oak Ridge Nat. Lab. Report ORNL-2309, Gatlinburg, Tennessee (1956) 56 14) J. M. Hammersley and D. C. Handscomb, Monte Carlo methods (Methuen, London, 1964) 15) W. Hauser and H. Feshbach, Phys. Rev. 87 (1952) 366 16) F. G. Perey, Phys. Rev. 131 (1963) 745 17) E. Gadioli, I. Iori, A. Marini and M. Sansoni, Nuovo Cim. 44B (1966) 338 18~ T. Ericson, Nucl. Phys. 11 (1959) 481 19) T. Ericson, Adv. in Phys. 9 (1960) 425 20) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1248 21) A. van der Woude, Nucl. Phys. 80 (1966) 14 22) I. I. Gurevich and M. I. Pevsner, Nucl. Phys. 2 (1957) 575 23) P. A. Moldauer, Phys. Rev. 171 (1968) 1164
(1970) 577--597; ( ~
North-Holland Publishing Co., Amsterdam
N o t to be reproduced by photoprint or microfilm without written permission from the publisher
I N T E R M E D I A T E STRUCTURE AND T H E REACTION
2SSi(p, p')ZSSi* A. C. SHOTTER, P. S. FISHER and D. K. SCOTT
Nuclear Physics Laboratory, Oxford Received 27 July 1970 Abstract: Angular distributions are reported for the reaction 2sSi(p, p')2sSi* measured over t h e incident energy range 12.0 to 15.0 MeV, at 100 keV intervals with a beam resolution of 100 keV. Total cross sections are determined for all residual states from 0.0 to 8.25 MeV. Cross-correlation analysis between the cross sections reveals structure similarities between the different excitation functions; this structure has a width of several hundred keV. An attempt is made to interpret the structure within the framework of the conventional statistical model. When the spacing of the compound levels is described by a Poisson distribution the model gives a fairly good representation of the data.
E I
NUCLEAR REACTIONS "sSi(p'P)' E=12"0-15"0 MeV; measured a(E;Ep;'O)"
]
29p deduced intermediate structure. Natural target.
1. Introduction In the past few years much interest has been focussed on the occurrence of intermediate structure in the excitation functions for nuclear cross sections and on the interpretation of the structure using the concept of doorway states 1,2). Although many experimental examples of intermediate structure have been reported, only one reaction channel is studied in many of these cases, and since it has been shown 3, 4) that the occurrence of intermediate structure is not alien to the statistical theory of nuclear reactions, some caution must be exercised when considering the interpretation of the data. In contrast to the conventional statistical theory, however, the doorway state theory predicts, in general, that similar intermediate structure should occur in all open channels 2). It is therefore of considerable interest to measure excitation functions corresponding to several inelastic channels, and then to study the correlation in structure between these functions. To this end, we have measured inelastic proton excitation functions for the nucleus 2SSi. This nucleus was chosen because it belongs to a mass region in which the total neutron cross sections, in the energy range of 4 to 12 MeV, undergo considerable fluctuations 5), and furthermore 28Si is unique in this mass region in having a doubly closed subshell, which would simplify calculations of the reaction mechanism. The reaction was investigated in the incident proton energy range 12 to 15 MeV, corresponding to an excitation in the compound nucleus of between 14.7 and 17.7 577
578
A.c.
SHOa'TER e t al.
MeV. Since'the purpose of the experiment was to search for intermediate structure, it was decided to reduce the amount of data to be analysed by measuring the reaction cross sections averaged over an incident energy interval of 100 keV. 2. Data collection and reduction The Van de Graaff accelerators of the Oxford Nuclear Physics Laboratory were used for this experiment 6). Details of the technique employed to measure energyaveraged cross sections are given elsewhere 7). Briefly, the method consisted of cyclically changing the incident beam energy by modulating the current energising the final analysing magnet in a saw-tooth manner with a period of 100 see. The deterioration of detector resolution that would have resulted from this modulation of the beam energy was avoided by electronically compensating the pulse-height spectrum of the inelastically scattered prgtons. In this way we found it possible to have a beam resolution of several hundred keV while maintaining an overall energy resolution between 25 and 35 keV in the inelastic proton spectra. ,
,
li:o
t
•
,
,
Peak. State (MeV)
0-0 Si
2 d
7
13 14
oo o
0.0 C
,
,
,
r
i
I
I
Peak. State (MeV) I I m I l I
16 18 19 20 11
8.16 Si 8.33 si 8"41Si 8'59 Si 8'908i 7.66C ~945
1.77 5J 4"61 Si 4"97 Sl 4"43C 6"1 7 5i 6"68 Si 6"88 S116-130 7.38,7.41 Si
I
22
9-17 Si
I [ i ~
23 14 28 26
9"31Si 9"dl 51 11"880 9'70 Si
6.920
I
27
7-80 Si;7.12 0 I 7" 93 5i
28 19
9.765i
9-93 Si 10"185i;9.590
J t.d Z Z
"1" tJ
0
400 CHANNEL Fig.
1. A
800
spectrum of scattered protons from the silicon target.
The angular distributions of the scattered protons were measured by simultaneously recording the pulses from four Si(Li) detectors mounted on a rotating table in a 60 cm diameter scattering chamber. The detectors, which had depletion depths of 2
2sSi(p, p')2SSi*
579
mm, were cooled to liquid nitrogen temperature to improve their resolution. The cross sections at sixteen angles could be obtained by taking data at four positions of the rotating table. The targets used for this experiment were self-supporting, 100/zg/cm 2 silicon films made by evaporation of natural silicon. A typical spectrum of scattered protons from such a silicon target is shown in fig. 1 and it can be seen that there is considerable carbon and oxygen contamination. During the course of the experiment about one thousand pulse-height spectra were collected; each spectrum consisted of one thousand channels. The spectra were analysed by a computer programme which included the following procedures: a search routine to determine the regions containing isolated or overlapping peaks; a routine to calculate the background, in regions containing peaks, by interpolating between peak-free regions; an iteration procedure to fit a Gaussian line shape to each peak located in the reduced spectra produced by subtracting the background; a kinematic routine to identify the peaks. The method of least-squares was used to fit each experimental angular distribution with an expression of the form do"
Im~z -
d12
-
~ BtP,(cos 0).
(1)
t=o
The integrated cross section was then calculated as 4roB0. 3. The experimental results The total cross sections for inelastic proton scattering to the first eleven states of zssi are shown in fig. 2. The experimental angular distributions change rapidly with energy, some indication of which can be appreciated from the plot of the Legendre coefficients in fig. 3. From fig. 2 it can be seen that the integrated cross sections exhibit a structure which has a width of several hundred keV; this structure is termed intermediate structure since its width is intermediate between the widths observed in "fluctuation" experiments and the potential resonances that are seen with beams of poor energy resolution. It is also apparent from fig. 2 that there is some similarity between the structures of the different excitation functions. In order to establish a quantitative level of significance for the similarity of any two excitation functions, cross-correlation functions were determined for all combinations of the cross sections. For any two cross sections tri(E), trj (E) the cross-correlation function, Rij (e), is defined by
Rij(e)
= ((o-,(E)-
where (a(E)> is the energy average of o-(E).
(2)
580
A.c. SHOTTER et aL
The cross-correlation functions for all combinations of the cross sections are shown in fig. 4. The uncertainties of R u (s), which are caused by the finite range of the experimental data, are represented by an error bar plotted at the side of each graph in Q = 0.0 HeY
40T
2000 l I0 O0t . -'% ~..,,,........,."" "%% ,,,",,~,,,...,..-"% 13 ENERGY
14 15 MEV o - 1.77 McV
z°° T/
/..,/'~ _ .,,%. .., lOOt ...,,..~,"~.o~",." ~ . . % . , ~ r - •
t IZ
z° T.''~ ,,,,,J ",,
12
" --|ER-5"/. I?.
14
15
,:
20T
tn Ul ~J Ul Ln ,( I:
B
',.j kj
I
13
t
IS
,"~/,,
z"~,
1
/ER- 3%
_o"
~'~
ER - 3"/0
lEE - 5% 12
=.~ ~
*"'-",../ %
Q - 7.38, 7-41 MeV .¢~,
I
I
14 15 ENERGY MEV Q- 7.8 MeV
,,,
....
I
I
J
13
14
15
~q,
,'=~
Q " 7.93 MeV
r v
";~..'"X
•
2O
ER-S%
/ ER" S %
12
13
20 t ,
14
f'~'~
¢'~
10 B" ' U , % ~
'~ .
Q'4'97MtV
~.,t,~'~.
"~/'
| ER=3%
IZ
I.. L
%¢
13
%' '
40 T
15
2oor
'.
,i ;
" ' , " ' ' % °-" SUM Q - 8'25 McV
/ ER= 4 %
I
I
I
12
13
14
i5
Q- 6.27 McV
L
'~ i
P
A
I00 SUM OF Q FROM 4.61 TO 7.93 McV
/ ER. ~./, 12
13
IO0T
14
15
Q= 6-88.6.89 McV
/
501"%%." /SR-n%
-',.'
V
13 LAB ENERGY
"
I
Iz
i,
1'4
Ik
4OOT%"
/
12
\j'v
,3
I00 'k ...~,,.,,e'.'
A
14
,
,2
15
'~
¢~
~ ' ~ J~t.
"~'%,.~'. 14
MeV
IS
| SUM OF QI FROM 1"77, TO 7.93 McV 12 13 14 15 LAB ENERGY MeV
Fig. 2. The integrated cross sections for the reaction 2ssi(p, p')Zssi*. The excitation energy o f the residual nucleus is denoted by Q, and the approximate percentage error on the experimental points is given by ER.
fig. 4. These errors were calculated from the formula derived by Hall 8), modified to take account of the finite energy resolution of the beam used in this experiment. If the different reaction cross sections are unrelated in energy, then in any one graph of fig. 4 there should be approximately the same number of points lying within the error bar as outside since the error bars correspond to the most probable errors. The
s SSi(p, P')2SSi*
"
o
" ~1 o
'
....
o
581
-, °1~
---
~r:2 "
'o
!.!
o
.g
L'.
~-.2
=
o= i
-~
c,
~
I
°~
~.~d, °
¢.,
o
'
o'
'o
~1~0
....
o
.
~
o
....
o
~1~
~,
.-'1,~
,~
,
o
-,,s-'l~
"6 .o
,i ,Q
b
,,,,.
i
o
ID
o.~., i
...,_J
..i
I:I
Z( ~
~
Z
•
|
g
cl
b IL
.S
o
~
~
'
,
.
;
qr
~1~
"~
0
i
ub
•
582
A.C. $HOTTER et aL
interpretation of the behaviour of R~s (s) for the graphs involving the excitation functions of the lower excited states of 28Si is complicated by the possible contribution of a direct component to the reaction mechanism; this will cause the value of R~s (~) E~ ~ATE ~
0"0 MeV
Ci I-~ ~V
4.61 MeV
~ 4.97 McV
• ..t_~o2
6.Z7 MeV
• .~L~;,
E~
E~
7.4 MeV
GSB MeV
7"8 MtV
t_~o,
7"9 McV
l -O'OZ
...+ • ..+ ÷ ..+.. + + . °.
•
+++ ....+..+ i
\
Fig. 4. The cross-correlation functions. Each c o l u m n consists o f graphs s h o w i n g the correlation between one particular reaction cross section and the other remaining cross sections. The cross sections are identified by the excitation energy o f the residual nucleus. The y-axis scale is the same for all graphs in any c o l u m n , and any t w o neighbouring points in any one graph are separated by e = 100 keV.
to be depressed as can be seen from eq. 2. Nevertheless, there are at least six different reaction channels for which the cross sections, when correlated with the cross section for the 1.77 MeV state, yield values of R, over an interval of 300 keV, greater than the errors due to the finite range of data.
2SSi(p, P')28Si*
583
Of the forty possible different combinations of the excitation functions only six cases have Rij (8) negative for 8 = 0. If the different reaction cross sections are unrelated in energy, and if the data are of infinite extent, then the value of R~j (8) would be zero. However a finite value of Rij (8) can arise due to the limited range of the data, and in such a case both positive and negative values of R~j (8) will occur with equal probability. The fact that so many experimental values of Rij (e = 0) are positive, and comparable in magnitude with the value of the autocorrelation function Rii(8 = 0), is indicative of some correlation between the reaction cross sections. If the states at 6.27 and 6.88 MeV, for which R~j fluctuates about zero, are excluded, the values of R~j (~) are positive for a region of several hundred keV centering around 8 = 0 and, for many of these cases, the values within this region are consistently greater than would arise by chance because of the finite range of data. A clear illustration of these correlations in the cross sections is seen in fig. 2, which in addition to showing the experimental excitation functions for individual states, shows the sum of cross sections for scattering to groups of states. It is clear that there is strong similarity between the structures of the single and summed excitation functions. 4. The statistical theory and intermediate structure The excitation functions shown in fig. 2 exhibit intermediate structure, and there appears to be some correlation between the structure for the different reactions. The question as to whether doorway states are responsible for the structure still remains to be answered. As noted by previous authors 3) statistical grouping of the compound nuclear parameters could be responsible for the structure. That this may indeed be so can be appreciated from the following simple argument. Consider a reaction which proceeds by a compound nuclear mechanism. At low bombarding energies discrete resonances are observed, whereas at higher bombarding energies the average energy spacing ( D ) between the resonances decreases rapidly, while the average width ( F ) of the resonances increases only slowly. For the energy region where there is strong overlapping between neighbouring resonances the cross sections become quite complicated, and for the region where (F)/(D)>> 1 the "resonances" or fluctuations in the cross section can no longer be identified with individual compound nuclear levels but are rather the result of interference between the amplitudes of many levels. The position and amplitude of these "resonances" are essentially random. Since many levels contribute to the reaction when (F)/(D) >> 1, and since the decay amplitudes to the different reactions channels are in general independent, the cross sections for different inelastic channels will in general be uncorrelated. This behaviour is to be contrasted with that in the lower energy region since in general an isolated resonance will decay to all open channels, and so there will be strong correlation between the different inelastic cross sections. In the intermediate energy region, where (F)/(D) ,~ 1, some correlation is still likely to exist between the inelastic cross sections.
A . c . SHOTTER et aL
584
For the experimental situation we have studied, (F)/(D) ~ 2. It has therefore been necessary to estimate the correlation between the reaction channels for this intermediate energy region before concluding that doorway states are the cause of structure correlations between different reaction cross sections. In the next section we present some details of the method we have used to calculate the statistical correlation when (F)/(D) ~ 1. Our notation and method are similar to those previously used by Ericson 9). 5. The statistical model
The cross section for a reaction between two states ~ and ~' may be written in the following form 10)
=
E(2t+
2.1+ 1
iS(aisl ,l,s,ljn)12'
(3)
1)(2i+ 1)
where S (adslat'l's'lJn) is the scattering matrix element describing the transition from the state • of channel spin s to a state r~' of channel spin s', through a compound system with a total angular momentum J and parity n, the incident particle having an orbital angular momentum 1, and the emergent particle an orbital angular momentum l'. The spin of the target nucleus i s / , and the spin of the incident particle is i; the sum is over the quantum numbers 1, l', s, s', J, n. It is assumed that the scattering matrix elements may be separated into two components, one associated with the compound nuclear mechanism and the other with a direct reaction mechanism 1i), i.e.
S(~lslat'l's'lJn) = ~ ~ - E , ~ - ~ +(S),
(4)
where the summation extends over all the compound nuclear levels associated with the quantum numbers J and r~; ( S ) is the energy-averaged component associated with a direct reaction mechanism; the term E~ = ei-½iF~, where ei and F~ are the energy position and width of the compound nuclear level i; the index/~ represents the quantum numbers/, s, 1', s'. For the purpose of this work it will be assumed that the real and imaginary parts o f the coe~cients a l ( ~ ' , / ~ J n ) belong to Gaussian distributions, and that the coefficients corresponding to a particular i, but to different values of (~a',/~Jn), are independent. With further assumptions about the distribution of the compound nuclear energy positions it is possible to calculate energy-averaged cross sections and the fluctuation from the mean. A measure of the local energy fluctuation of a cross section is given by the variance o f the cross section defined by var (a) = =
((a-(a)) z)
where the angle brackets denote an average over energy.
(5)
2aSi(p,P')iSSi*
585
For simplicity the direct reaction component of the scattering matrix is assumed to be zero; in this case
a(~l~')
ai(~tot',flJn)a*(otot',flJn) g(J) ~', .h,ts U (E- Ei(Jn))(E- E~(Jn))
= Y.
t6)
where
g(j)
=
nA2(2J+l)
(21+0(2i+0 .Using this equation the averaged square of the cross section may be written
<~2(=1=')> = E g(d)g(J') ,lx# ,,l'n'B'
a,(flJn)a~([JJ~c)am([3'J'rg)a*(fl'J'n') \ × ,~. (E- Ei(Jrc))(E- Ej(Jrc))*(E- E.(J n'))(E- E.(J'n'))*I"
(7)
When the random nature of the coefficients is considered, the sum of the terms in this expression for which J = J', n = n', fl = fl' becomes Eg2(J) I2 ( ~
.r~p
fa'[2lajl2\ I(E-~Ej)I2/,,j
ie-e,i /
\,, ( e - ~ E * ) ' / , + , / "
(s)
The values of the first and last terms in this expression depend upon the correlation between the energies of neighbouring compound nuclear levels. If the levels of particular J and n are randomly distributed along the real energy axis, then expression (8) becomes x2) la'14 . \ l E F)
E I2
(9)
where D(Jn) is the average level energy spacing for states with the quantum numbers J and n, and F(Jn) is the average total width for the compound levels.
A . C . SHOTTER e t al.
586
By using "similar procedures to those just outlined, the average cross section may be written (a(~[~')) = ~ F ( ~ n ) ( l a ' 1 2 ) 2
(10)
and so expression (9) becomes )-" 2(a(~l~'))s2.p[1
+O(Jn)/nF(Jn)].
(11)
drip
The evaluation of the terms of eq. (7) for which one of the equalities J = J', n = n', p = ~' does not hold, follows along similar lines as the evaluation of expression (11 ); the result is
'
'
F hpp,6ss, 6.., [_nF(Jn)/D(Jn)
(a(~[~))s.p(a(~[~ ))s'.'p'
]
+1 ,
(12)
J,x,p,
where hpp, = I if/~ #/V, hpp, = 0 if p = p', and 6 is the Kronecker symbol. After collecting terms for eq. (5), the variance of the cross section becomes i
var (a(~[~')) = Z (a(~l~'))s2.p [1 + D(Jn)1 s.p ~F(Jn)]
2
~r(Jn)/D(Jn)
(13)
When the cross section includes scattering to several residual states, so that a=Za(~[~,) ,
(14)
n
the variance can be evaluated by similar methods to those outlined above, with the result var (a) = E (a(~l~.))~s. (1+ D(Jn) ~ .ps. \ ~r(Jn)/
+
cr(~[~n)#sg)2 #n
nr(Jn)/D(J~)
(15)
6. Numerical calculations The validity of eq. (15) was tested by simulating excitation functions from preset parameter distributions of the energies and the amplitudes of the compound states. The starting point of these calculations was the S-matrix
S(E)
= ~
a,
E~-E~
= ~
~,+i]~,
• E-Ei+½iF'
(16)
where it is assumed that the real and imaginary parts of the reaction amplitude as belong to Gaussian distributions of the same dispersion, but are otherwise unrelated.
2asi(p, p')2aSi*
587
We investigated excitation functions characterized by three types of level spacing distributions; constant level spacing, a Poisson distribution of level spacing and a Wigner 13) distribution of level spacing which corresponds to a level repulsion. The random numbers used for these calculations were generated as a pseudorandom sequence 14) by the congruential method using a module of 515. The excitation functions were generated by calculating the expression tr(E) oc Y'
(a,+ ifl,)(~j+ iflj)*
(17)
77( E - E, + ½ir)(E- E j - ½iF) as a function of E. At each energy E, the summation in this equation should have extended over all resonances i, but in practice only states within an energy interval of the order F made any significant contribution. For the calculations to data we have used a summation interval of 10F, i.e. for a particular value of E only those levels which occur in the interval E - 5 F to E + 5F are considered. TABLE I Variance o f artificial excitation functions <_P)/
Average n u m e r i c a l variance
Theoretical variance
c o n s t a n t spacing
0.126
l/n = ~- = 0.125; fluctuation theory
r a n d o m spacing
0.390
0.4; calculated from eq. (15)
W i g n e r spacing
0.240
The ratio of the terms occurring on the right-hand side of eq. (15) depends upon the number of reaction channels n. Therefore in order to enhance the contribution of the second term we have studied excitation functions comprising eight independent channels. Since these calculations require considerable computing time, we have so far considered only excitation functions parameterized by F/D = 2, which is closest to our experimental situation. The average numerical variances for the excitation functions associated with the three types of level distributions mentioned above are shown in table 1, together with the value predicted from conventional fluctuation theory. It is seen that the variance for excitation functions characterized by a Poisson distribution is about three times greater than the value given by fluctuation theory, but, as shown in this table, the variance is in reasonable agreement with the value calculated from eq. (15). As expected, the variance of the excitation functions characterized by a Wigner spacing is substantially smaller than the value for the case of Poisson spacing, while the variance of excitation functions with constant level spacing agrees with the value predicted by fluctuation theory 9).
:588
A.C.
SHO'I'rER
et
ai.
oo Q
•
°~
°~o
°
o. °
° ° oo
°o
.
:°
oo
.eo
~°
* %
. • o°°
*~oo°
** .o
° o U
oo
%
°o
I
J~*
o o
°'-
._
°000•
0
..
0
°o °
.
Q~
oe
-.
i
°~
4.J
~o
00 ..
•o * o
$
:.
%
~0 o
.° ° o
0
**~
°o °
°o
-.
!
:.
oo
~o , ....
|
~° °,, oo
°° •
.,° /%
°o e
°o ° o
N O IJ. ~ 3 $ - - SSOkl;]
~ V
:naJ(p, p')2esi*
389
m
0o •
° oo°
~
o 0% e
1
,,'°
.':
oe
|.
oO_° •
$
On
o" oi
o| •
e|
Oo
|
o o
I
o o
"'i ".
#|
• -....
"8
i""
.i 8
o o
4,o
oo
0=
°8|
8
.5 0i 6°
:".',
.J
°o •
8 t • • |
OI ID
eee •
7° • i0"
ii ¢:1
.'o %
~o
me
OQ° | e°
~.~ e
X
Oo
%
.:,o L. NOIIO3S--SSO~d~
° o
590
A . C . SHOTI*ER e t al.
The grea~er variance of the simulated excitation functions for random level spacing compared with constant level spacing is primarily due to a grouping of the levels, which produces similarities of structure in the different excitation functions. This point is illustrated in fig. 5 where three different excitation functions belonging to the same random sequence of energy levels are shown; there is an obvious similarity between the curves. The three excitation functions displayed in fig. 6 correspond to a constant level spacing, and in contrast to fig. 5, there is no similarity between these curves. 7. The experimental results and the statistical model In this section we outline the method and approximations used to evaluate the cross-section variances from eq. (15) and we then compare these calculated values with the experimental variances. The average cross sections appearing in eq. (I 5) for the Variance of the crosssections are for specific values of J and ~ in the compound system. These cross sections were calculated using the formalism of Hauser and Feshbach 15). The relevant transmission coefficients, which are required by the formalism, were taken from the work of Percy 16) and the ratios F(JTt)/D(Jn) were calculated by means of the statistical model 17) which gives 1
F(Jz)/D(Jn) = ~ ~
Q(IJrOf p(E*,)T~(E,)dE,,
(18)
where Q(lJn) = ~f(jTt), a n d f ( j ~ ) is the spin distribution function for the residual nucleus. In this equation r denotes a particular decay mode e.g. p or 'tHe, p(E*) represents the total density of levels for the appropriate residual nucleus at excitation energy E*. The transmission coefficient for an emitted particle of spin I decaying from a compound nuclear state with quantum numbers J~, with orbital angular momentum l, and energy E,, is denoted by 7", (E~). To simplify the evaluation of eq. (18) the dependence of p upon E was taken to be 18) p = exp (E-E°-----~)
(19)
7"0 and the spin distribution function to be 19) J (jrt) = (2j + 1) exp ( - (j + ½)2/2a2). 2tr 2
(20)
The values of the constants Eo, To and tr were taken from the work of Gilbert and Cameron 20). When the average cross sections, determined by means of the above formula, were summed with respect to J and 7t the values were less than the corresponding average experimental cross sections by a factor of 1.6. This underestimation is probably due to an incorrect estimation of the absolute magnitudes of the ratios F(Jn)/D(Jn) calculated from eq. (18), and so, in subsequent work, the calculated average cross sections were multiplied by a factor of 1.6.
2Ssi(p, p')28Si*
591
The variances of the cross sections calculated from eq. (15) were multiplied by a damping factor to take account of the finite energy spread of the incident proton beam; this factor was deduced from the work of Van der Woude 21). The calculated variances are compared with the experimental values in fig. 7; it is seen that within the experimental errors there is approximate agreement between theory and experiment.
/
8O E A
6C
I
I
STATES I. Q " 1.77 MeV 2. Q'4.61 McV 3. Q "4"97 McV 4. Q - 6.27 McV 5. Q " 6'88+6.89 MeV 6. Q-7-38+7.41 McV 7. Q -7.80 McV 8. 0 "7'93 MeV
g
2C
I
I
I
I
20
I
40
I
I
I
I
I
60 80 I00 EXPERIMENTAL VAR (o) mb2
I
I
I
120
I
140
Fig. 7. The calculated and experimental variances of the integrated cross sections for the reaction 28Si(p, p')28Si*. The different cross sections are identified by the corresponding energy of the residual nucleus. The line represents the equation y = x.
In our investigation of the correlation in structure between the experimental excitation functions we have found it more advantageous to study the variance of excitation functions produced by compounding several of the experimental functions, rather than to study the correlation between the individual functions. The reason for this may be appreciated from the following argument. If we suppose that the inelastic cross sections are uncorrelated in energy, then the variance of a compounded excitation function should be given by the equation var
(E a(~l~.)) = E var (a(~]=.)), n
(21)
n
where n labels a particular reaction channel. If, however, eq. (15) is correct eq. (21) must be incorrect, and in this case should be replaced by
(E o(=l=.),p>(E var
(E o(~1~.)) = ~ var (o(~l~.)) + Z n
n
Jxlln'
P
#
nl"(J=)/O(Jg)
(22)
From this formula it is evident that the ratio of the second to the first term is proportional to n, the number of channels, and so structure correlations between the indi-
592
A.C. SHOTTER et al.
vidual excitation functions become more apparent in compounded functions. It is worth noting that eq. (22) reduces to eq. (21) as the ratio F(J~)/D(JTr)tends to infinity. In fig. 8 the variances of several experimental compounded excitation functions are plotted against the values calculated from eq. 21; it is evident that the experimental values are greater than the values predicted from eq. (21) - a result indicative of d(R) SUM OF CROSS SECTIONS FOR / STATES WITH Q VALUES: / 4-61, 4.97, &Z7, 6.88+, 7.41",7 - ~
Nt 3°°
/
.~ Z00
•
7 :
• • Z. . . . 3. 4. 5.-
/ c~
/ IOC
,,f
/
/
6.-
7. ,,
-vor(o (R)-o (dF61)) vor(d (R) -o (6"27)) = mr(¢ (R) -o (6.88)) - vor(o (R)-o (7"4)) - vor (o (R)-el n'-B)) - w r (e (R)-e (T.9)) - vor(o (R))
200 300 i 400 EXPERIMENTAL VAR (~) mb2
t 500
Fig. 8. T h e calculated a n d experimental variances o f several c o m p o u n d e d cross sections for t h e reaction zsSi(p, p')ZsSi*. T h e n u m b e r n o t a t i o n identifies different c o m b i n a t i o n s o f the experimental cross sections. T h e calculated variances were evaluated u s i n g eq. (21). T h e line represents t h e e q u a t i o n y = x. .j
7
400 ! 3O0
~ZOO
i
NUMBER NOT AS FOR FIGAsTION
I00 /
'
300 400 EXPERIMENTAL VAR (or) rnbz
500
Fig. 9. The calculated and experimental variances of the compounded cross sections for the reaction zsSi(p, p')ZsSi*. T h e calculated values were evaluated using eq. (22). T h e line represents the equation y ~ .*¢.
2SSi(p, P')28Si*
593
structure correlation. Fig. 9 is similar to fig. 8 except that the calculated values were derived from eq. (22); clearly, for this case, there is better agreement between the calculated and experimental values. The next two sections deal with the experimental angular distributions, and in particular with their dependence upon energy. 8. The Legendre coefficients The angular distributions may be parameterised by a set of coefficients BL, defined by the equation [ref. lo)] do - ~ B,(ulu')Pr.(cos 0). dO L
(23)
The coefficients B, are related to the scattering matrix by the equation X2 BL(Ctl*t') = 4(2• + 1)(2i + 1) ~ ( - 1)"-SZ(ll J112 J2; × Re
sL)Z(l~ Jl 1'2J2; s'L)
[S(*tll sl~t'l l't s'lJx nx)S*(ot/2 slot'l'2s'[J27t2)'] ~t2
4(2• + 1)(2/+ l) L~ + E w(cl, c2, L) Re
~(c,
¢,
L)S~,(c)
[S,=,(cl)S*,(c2)]],
(24)
ClC2
cl -~c2
where cl is an abbreviation for the quantum numbers 11, I~, s, s', Jx, n~ and = (ct, c2) = (-1)~'-*Z(ltJll2J2; sL)Z(I~Jll~J2; s'L); otherwise the notation is the same as for eq. (3). The first term in eq. (24) is non-zero only for even L coefficients; the second term may be separated into two components, depending on (cx, c2), one associated with odd coefficients, the other with even coefficients. Since the same combinations of (ct, c2) appear in all the coefficients with L even, it is probable that these coefficients will exhibit some correlation. Similar conclusions hold for the coefficients with L odd. It has been further shown 9) that, if the direct component of the scattering matrix is zero and if the compound nuclear component is completely independent for the different channels c, then there is zero cross-correlation between the odd and even coefficients BL. The experimental values of the coefficients BL, which are shown in fig. 3, are not inconsistent with these conclusions, since there is some evidence of correlation between the coefficients with L even and also between the coefficients with L odd. The effect is particularly marked for inelastic scattering to the 4.97 MeV state of 2SSi; furthermore for this state there is little or no correlation between the even and odd L coefficients.
594
A . c . SHOa'rER et al.
9. Variance of Legendre coefficients By using the same assumptions that were employed in the derivation of eq. (15) it can be shown ,2) that the variance of the coefficient BL is related to the partial reaction cross sections a(flJn) in the following manner var (BL(Ula')) = ~ w2(c, c, L)(a(c))2 + ~ w2(cl , c2, L)(a(Cl))((7(c2)) Cl, ¢2 C l ~= C2
C
1
+ c,~',c, nF(Jrc)/O(Jn)
[w(cx, cx, L)w(c2, c2, L)+w2(c,, c2, L)]
where c = (fl, J, n), fl = (l, s, l', s') and Y" denotes a summation with the restriction J, = J2 = J, rq = n2 = re. The coefficient w(cD cz, L) is defined by
w(c,, c2, L) = z'(ll, a,, h,
s , , t;, s,; s'L)
4n~/(2J 1 + 1)(ZJ2 + 1) Comparison of the terms in eq. (25) with those of eq. (15), in association with eq. (24), suggests they are similar in origin. The first term of eq. (25) is analogous to the first term of eq. (15) and arises mainly from interference between the compound resonances associated with a particular set of quantum numbers c. The second term in eq. (25) arises from an interference between compound resonances associated with different sets of the quantum numbers c; this interference is also represented by the second term in eq. (24). The third term in eq. (25) is analogous to the second term of eq. (15), and arises from the contribution of the single resonance states to the variance; this term becomes negligible when F/D >> 1. TABLE 2 Value o f term in eq. (25) (mb/sr) 2 L
1st term
1 2
3rd term
95 29
3 4
2nd term
104
7
134 14.2
124
3.3
We have applied eq. (25) specifically to the case of inelastic scattering to the 4.97 MeV state of 2aSi, which has spin 0 and therefore makes the evaluation simple. The calculated values of the terms occurring in eq. 25 for this reaction channel are tabulated in table 2. It is clearly seen that the value of the second term, due to interference between levels with different J and n, dominates the other terms. The calculated and experimental values of the variances of the coefficients BL are plotted in fig. 10. There
2SSi(p ' p,)ZSSi*
595
is reasonable agreement, although the large experimental errors make it difficult to draw any firm conclusions. 4C ;
.(3
•
l
I
3c ..J n,-
i
•
IL VALUES 0. I. 2. 3. 4.
J
~ 1c J < U
I
I I 20 EXPERIMENTAL
I I I 40 6O VAR (B L) ( m b / s r ) 2
L'0 L-I L-2 L-] L'4 I
I
80
Fig. 10. The calculated and experimental values of the variances for the Legendre coefficientsof the reaction zssi(p, p')Zssi*, Q = 4.97 MeV. The line represents the equation y = x.
10. Discussion and conclusions The motivation for the experiment outlined in this paper was the proposal that intermediate structure may occur in reaction cross sections, and that the structure associated with different reaction channels may be correlated with energy. We have measured excitation functions for several channels of the reaction ~aSi(p, p')2aSi*, and we have used the cross-correlation function to determine if there is any similarity between these functions. The results of these studies give evidence for structure correlation between the excitation functions, which in the past has been taken as convincing evidence for the existence of doorway states. However, before identifying this intermediate structure with doorway states we have studied the conventional statistical model in order to see if this can account for our experimental results, in particular for the similarity of excitation functions for different reaction channels. The picture of intermediate structure that emerges from the statistical model is that of a grouping or clustering of compound nuclear levels at particular energies; the grouping arises because the energy positions of the compound nuclear levels are essentially random. This grouping of the compound nuclear levels does of course influence the cross sections for all open channels, and so tends to produce similar structure in all reaction cross sections. The degree of correlation between cross sections depends upon the magnitudes of the ratios F(J:t)/D(Jn); for values much greater than unity the correlation will be very small, but, for values around our experimental value of 2, the correlation becomes significant. If it is assumed that the energy positions of the compound levels are described by a Poisson distribution along the energy axis, then the grouping of the compound levels gives rise to correlation similar to that ob-
596
^. c. SHOTTERet aL
served experimentally. If, however, the distribution of the le,vel spacing is of the Wigner type 13), then the levels will be more evenly spaced than in the previous case, the grouping of the levels will be less pronounced, and so the correlation between reaction excitation functions will be smaller. The results of the numerical calculations discussed in sect. 6 lead to the conclusion that if there is a Wigner distribution then the grouping of the compound levels will be too insignificant to explain our experimental results. There is considerable experimental evidence 22) to show that the level spacing distribution at low excitation energies is of the type proposed by Wigner. At higher energies where the levels overlap the situation is far from clear, but it is usually assumed that the mathematical form of the level distribution is similar to that for low energies. However, recent calculations by Moldauer 23) suggest that this may not always be true. In these calculations resonance pole parameters of the scattering matrix are calculated from statistical models of the K-matrix, so ensuring the unitary character of the scattering matrix. The results of these calculations clearly show that the repulsion between levels decreases when the channel transmission coefficients and the number of open channels increase. F o r the experimental situation we have studied, several of the transmission coefficients are in fact quite large and the number of open channels is high, and so from Moldauer's results it might be concluded that the compound levels of particular J and n show little repulsion. I f this is true, then the levels are randomly spaced, and this is exactly what is needed to support the interpretation that the experimental structure correlation is largely a statistical effect. Further experimental data are clearly required before forming definite conclusions about the observed intermediate structure. One possible experiment would be similar to the one set out in this paper but for a heavier target nucleus or a higher bombarding energy so that F ( J n ) / D ( J n ) >> 1. F o r this case a conventional statistical interpretation of any observed similarity between the excitation functions would be untenable. We are grateful to Professor D. H. Wilkinson and Professor K. W. Allen for the use of the Oxford electrostatic generators. We wish to thank the technical staff of this laboratory for their willing support. A.C.S. and D. K. S. wish to thank the Science Research Council for financial support.
References 1) H. Feshbach, Int. Conf. on the study of nuclear structure with neutrons, ed. M. N~ve de MCverghies et aL (North-Holland, Amsterdam, 1966) 257 2) H. Feshbach, A. K. Kerman and R. H. Lemmer, Ann. of Phys. 41 (1967) 230 3) P. P. Singh, P. Hoffman-Pinther and D. W. Lang, Phys. Lett. 23 (1966) 255 4) W. R. Gibbs, Phys. Rev. 181 (1969) 1414 5) K. Tsukada and O. Tanaka, L Phys. Soc. Jap. 18 (1963) 610; U. Fasoli, D. Toniolo, G. Zago and F. Fabriani, Nuovo Cim. 44B (1966) 455; A. D. Carlson and H. H. Barschall, Int. Conf. on the study of nuclear structure with neutrons, ed. M. NCve de MCvergnies et aL (North-Holland, Amsterdam, 1966) 537
zsSi(P, p')2s$i* 6) 7) 8) 9) 10)
597
W. D. Allen and R. H. V. M. Dawton, Rutherford lab. report RHEL/R/120 (1966) A. C. Shorter, J. Takacs and P. S. Fisher, Nucl. Instr. to be published I. Hall, Phys. Lett. 10 (1964) 199 T. Ericson, Ann. of Phys. 23 (1963) 390 H. Feshbach, Nuclear spectroscopy, Part B, ed. F. Ajzenberg-Selove (Academic Press, New York and London, 1960) 625 11) H. Feshbach, Ann. of Phys. 43 (1967) 410 12) A. C. Shorter, D.Phil. thesis, University of Oxford (1968) 13) E. P. Wigner, Oak Ridge Nat. Lab. Report ORNL-2309, Gatlinburg, Tennessee (1956) 56 14) J. M. Hammersley and D. C. Handscomb, Monte Carlo methods (Methuen, London, 1964) 15) W. Hauser and H. Feshbach, Phys. Rev. 87 (1952) 366 16) F. G. Perey, Phys. Rev. 131 (1963) 745 17) E. Gadioli, I. Iori, A. Marini and M. Sansoni, Nuovo Cim. 44B (1966) 338 18~ T. Ericson, Nucl. Phys. 11 (1959) 481 19) T. Ericson, Adv. in Phys. 9 (1960) 425 20) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1248 21) A. van der Woude, Nucl. Phys. 80 (1966) 14 22) I. I. Gurevich and M. I. Pevsner, Nucl. Phys. 2 (1957) 575 23) P. A. Moldauer, Phys. Rev. 171 (1968) 1164